Definable Lipschitz selections for affine-set valued maps

Abstract: Whitney's extension problem, i.e., how one can tell whether a function $f : X \to \mathbb R$, $X \subseteq \mathbb R^n$, is the restriction of a $C^m$-function on $\mathbb R^n$, was solved in full generality by Charles Fefferman in 2006. In this paper, we settle the $C^{1,\omega}$-case of a related conjecture: given that $f$ is semialgebraic and $\omega$ is a semialgebraic modulus of continuity, if $f$ is the restriction of a $C^{1,\omega}$-function then it is the restriction of a semialgebraic $C^{1,\omega}$-function. We work in the more general setting of sets that are definable in an o-minimial expansion of the real field. An ingenious argument of Brudnyi and Shvartsman relates the existence of $C^{1,\omega}$-extensions to the existence of Lipschitz selections of certain affine-set valued maps. We show that if a definable affine-set valued map has Lipschitz selections then it also has definable Lipschitz selections. In particular, we obtain a Lipschitz solution (more generally, $\omega$-H\"older solution, for any definable modulus of continuity $\omega$) of the definable Brenner-Epstein-Hochster-Koll\'ar problem. In most of our results we have control over the respective (semi)norms.